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Symmetry and Introduction to Group Theory

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Solid State Theory Physics 545 Symmetry and Introduction to Group Theory StillddifdtltftSymmetry is all around us and is a fundamental property of nature. Syyymmetry and Introduction to Group py Theory The term symmetry is derived from the Greek word “symmetria” which means “measured together”. An object is symmetric if one part (e.g. one side) of it is the same* as all of the other parts. You know intuitively if somethinggy is symmetric but we req uire a precise method to describe how an object or molecule is symmetric. Group theory is a very powerful mathematical tool that allows us to rationalize and simppylify many yp problems in Chemistr y. A grou p consists of a set of symmetry elements (and associated symmetry operations) that completely describe the symmetry of an object. We will use some aspects of group theory to help us understand the bonding and spectroscopic features of molecules. Space transformations. Translation symmetry. " Crystallography is largely based Group Theory (symmetry). " Symmetry operations transform space into itself. Simplest symmetry operator is unity operator(=does nothing). (=Lattice is invariant with respect to symmetry operations) " Translation operator, TR, replaces radius vector of every point, r, by r’=r+R. Symmetry a state in which parts on opposite sides of a plane, line, or point display arrangements that are related to one another via a symmetry operation such as translation, rotation, reflection or inversion. Divided into two ppyarts by a plane Mirror plane (symmetry element) Have no symmetry remaining : Asymmetric units AtiAsymmetric unit Recall that the unit cell of a crystal is the smallest 3-D geometric figure that can be stacked without rotation to form the lattice. The asymmetric unit is the smallest part of a crystal structure from which the complete structure can be built using space group symmetry. The asymmetric unit may consist of only a part of a molecule, or it can contain more than one molecule, if the molecules not related by symmetry. Protein Crystal Contacts by Eric Martz , April 2001 . http://molvis.sdsc.edu/protexpl/xtlcon.htm Group Theory mathematical method a molecule’s symmetry information about its properties (i.e ., structure , spectra , polarity, chirality, etc… ) A-B-A is different from A-A-B. O C However, important aspects of the symmetry of H2O and CF2Cl2 are the same. Symmetry and Point Groups Point groups have symmetry about a single point at the center of mass of the system. Symmetry elements are geometric entities about which a symmetry operation can be performed. In a point group, all symmetry elements must pass through the center of mass (the point). A symmetry operation is the action that produces an object identical to the initial object. The symmetry elements and related operations that we will find in molecules are: Element Operation Rotation axis, Cn n-fold rotation Improper rotation axis, Sn n-fold improper rotation Plane of symmetry, σ Reflection Center of syyymmetry, i Inversion Identity, E The Identity operation does nothing to the object – it is necessary for mathematical completeness, as we will see later. There are two naming systems commonly used in describing symmetry elements 1The1. The Schoenflies notation used extensively by spectroscopists 2. The Hermann-Mauguin or international notation preferred by cystallographers Symmetry Operations/Elements A molecule or object is said to possess a particular operation if that operation when applied leaves the molecule unchanged. Each operation is performed relative to a point, line, or plane - called a symmetry element. There are 5 kinds of operations 1. Iden tity 2. n-Fold Rotations 3. Reflection 4. Inversion 5. Improper n-Fold Rotation The 5 Symmetry Elements Identity (C1 α E or 1) Rotation by 360° around anyyy randomly chosen axes throug h the molecule returns it to original position. Any direction in any object is a C1 axis; rotation by 360 ° restores original orientation. Schoenflies : C is the syygmbol given to a rotational axis 1 indicates rotation by 360° Hermann-Mauguin : 1 for 1-fold rotation Operation: act of rotating molecule through 360° Element: axis of symmetry (i .e . the rotation axis) . Symmet ry Eleme nts Rotation turns all the points in the asymmetric unit around one point, the center of rottitation. A rot ttiation d oes not ch ange the handedness of figures in the plane. The center of rotation is the only invariant point (point that maps onto itself). Good introductory symmetry websites http://mathforum.org/sum95/suzanne/symsusan.html http://www.ucs.mun.ca/~mathed/Geometry/Transformations/symmetry.html Rotation axes (Cn or n) Rotations through angles other than 360°. Operation: act of rotation Element: rotation axis Symbol for a symmetry element for which the operation is a rotation of 360°/n C2 = 180° , C3 = 120° , C4 = 90° , C5 = 72° , C6 = 60° rotation, etc. Schoenflies : Cn Hermann–Mauguin : n. The molecule has an n-fold axis of symmetry. An objjypect may possess several rotation axes and in such a case the one (or more) with the greatest value of n is called the principal axes of rotation. n-fold rotation - a rotation of 360°/n about the Cn axis (n = 1 to ∞) O(1) 180° O(1) H(2) H(3) H(3) H(2) In water there is a C2 axis so we can perform a 2-fold (180°) rotation to get the identical arrangement of atoms. H(3) H(4) H(2) 120° 120° N(1) N(1) N(1) H(4) H(2) H(4) H(3) H(2) H(3) In ammonia there is a C3 axis so we can perform 3-fold (120°) rotations to get identical arrangement of atoms. The Principal axis in an objjgect is the highest order rotation axis. It is usually easy to identify the principle axis and this is typically assigned to the z-axis if we are using Cartesian coordinates. Ethane, C2H6 Benzene, C6H6 The principal axis is the three-fold The principal axis is the six-fold axis axis containing the C-C bond. througggh the center of the ring. The principal axis in a tetrahedron is a three-fold axis going through one vertex and the center of the object. Symmet ry Eleme nts Reflection flips all points in the asymmetric unit over a line, which is called the mirror , and thereby changes the handedness of any figures in the asymmetric unit. The poi nt s al ong th e mi rror li ne are all invariant points (points that map onto themselves) under a reflection. Mirror p lanes ( ⌠ or m) Mirror reflection through a plane. Operation: act of reflection Element: mirror plane Schoenflies notation: Horizontal mirror plane ( ⌠h) : plane perpendicular to the principal rotation axis Vertical mirror plane ( ⌠v) : plane includes principal rotation axis Diagonal mirror pp(lane ( ⌠d))p : plane includes princi pal rotation axis and bisects the angle between a pair of 2-fold rotation axes that are normal (perpendicular) to principal rotation axis. ShSchoen flies : ⌠h, ⌠v, ⌠d Hermann–Mauguin : m Reflection across a plane of symmetry, σ (mirror plane) O(1) σv O(1) H(2) H(3) H(3) H(2) These mirror planes are called “vertical” mirror planes, σv, because they contain the principal axis. O(1) σv O(1) The reflection illustrated in the top diagram is through a mirror plane H(2) H(3) H(2) H(3) perpendicular to the plane of the water molecule. The plane shown on the bottom is in the same plane as the Handedness is changed by reflection! water molecule. Notes about reflection operations: - A reflection operation exchanges one half of the object with the reflection of the other half. - Reflection ppylanes may be vertical, horizontal or dihedral (more on σd later). - Two successive reflections are equivalent to the identity operation (nothing is moved). σ h A “horizontal” mirror plane, σh,is, is perpendicular to the principal axis. This must be the xy-plane if the z- axis is the ppprincipal axis. σ σ In benzene, the σh is in the plane d d of the molecule – it “reflects” each atom onto itself. σh Vertical and dihedral mirror planes of geometric shapes. σv σv Symmet ry Eleme nts Inversion every point on one side of a center of symmetry has a siilimilar poiitnt at an equal distance on the opposite side of the center of symmetry. CtCentres of fi inversi on ( cent re of symmet ry, i or 1 ) Operation: inversion through this point Element: point Straight line drawn through centre of inversion from any point of the molecule will meet an equi val ent poi nt at an equal di st ance b eyond th e cent er. Schoenflies : i Hermann–Mauguin : 1 Molecules possessing a center of inversion are centrosymmetric Inversion and centers of symmetry, i (inversion centers) In this operation, every part of the object is reflected through the inversion center, which must be at the center of mass of the object. 1 F Cl 22F Cl 1 2 Br i Br 1 1 221 12Br Br 1 Cl F22Cl F1 i [x, y, z] [-x, -y, -z] We w ill no t cons ider the ma tr ix approac h to eac h o f the symme try opera tions in this course but it is particularly helpful for understanding what the inversion operation does. The inversion operation takes a point or object at [x, y, z] to [-x, -y, -z]. Aftii(iAxes of rotary inversion (improper rot ttiSation Sn or n ) Involves a combination or elements and operations Schoenflies : Sn Element: axis of rotatory reflection The operation is a combination of rotation by 360°/n (Cn) followed by reflection in a plane normal ( ⌠h) to the Sn axis.
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